Lect12_Wavepacket

# Lect12_Wavepacket - Momentum vs Wavevector Instead of...

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Momentum vs. Wavevector Instead of momentum, it is often convenient to use wavevector states: h P K = k k k K = ) ( k k k k = δ π 2 x k i e k x = 1 = + k k dk [ ] i K X = , [ ] 0 , = K P k k k k k k p p p p = = = = h h h h 1 ) ( 1 ) ( ) ( δ δ δ k = h p p = h k Wavevector definition: Important commutators: Eigenvalue equation: Relation to momentum eigenstates: k p p c k h = = Normalization Closure Wavefunction

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Wavefunction in K-space What is the wavefunction in K-basis? ) ( ) ( ) , ( 0 2 0 2 2 0 2 0 k k e k k e t k t k m t k i m t k i = = = δ ψ ψ h h k k 0 ) , ( t k ψ 0 ) 0 ( k = ψ Let 0 2 2 0 ) ( k e t t M k i h = ψ Then:
Quantum mechanical kinetic-energy eigenstate: Only Global phase is changing in time Global phase can’t be observed Is anything actually moving? The spatial phase-pattern is moving, but at: This is called the `phase velocity’ Not related to particle velocity in classical limit Classical free-particle motion: How can we have a `classical limit’ of QM if nothing moves at v=p /M? Wavefunction in X-space 0 0 ) ( ) ( p t p t v x t x = + = 0 2 2 0 ) ( p e t t M p i h = ψ h h π ψ 2 ) , ( 2 0 = t M p x p i e t x M p v 2 0 = h π ψ 2 1 ) , ( 2 = t x

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`Motion in QM The answer is that in QM motion is in interference effect Consider a quantum-superposition of two plane-waves: The interference pattern can move! + = 0 2 0 2 2 2 1 ) ( 2 0 2 0 k e k e t t M k i t M k i h h ψ π π ψ 2 2 ) , ( 0 0 0 0 2 2 + = t M k x k i t M k x ik e e t x h h
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